[PPT] - Fun With Number Systems Or: How Knowing Number Systems Can Help You PowerPoint Presentation

SLIDE 1

Fun With Number Systems

Or: How Knowing Number Systems Can Help You Interview Better

SLIDE 2

Finding the Odd Ball

SLIDE 3

SLIDE 4

9 10 11 1 2 3

SLIDE 5

9 10 11 1 2 3

SLIDE 6

Goal: Find the odd ball and whether it's heavier or lighter in three weighings.

SLIDE 7

The Solution

5 6 7 11 8 9 10 12 2 3 4 11 5 6 7 12 1 4 7 8 2 5 10 11

SLIDE 8

Balanced Ternary

Number system for encoding three-way

comparisons.

Each digit corresponds to a power of three.
Digits are -1, 0, +1.
For notational simplicity, will use -, 0, +.
Example: +0-0
1 x 33 + 0 x 32 – 1 x 31 + 0 x 30 = 24
Example: --++
-1 x 33 – 1 x 32 + 1 x 31 + 1 x 30 = -32

SLIDE 9

12 to +12 in Balanced Ternary

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

000

SLIDE 10

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

SLIDE 11

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

SLIDE 12

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

SLIDE 13

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10

SLIDE 14

1 2 3 4 5 6 7 8 9 10 11 12

1 4 7 10

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

SLIDE 15

1 2 3 4 5 6 7 8 9 10 11 12

1 4 7 10

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

2 5 8 11

SLIDE 16

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

SLIDE 17

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

SLIDE 18

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

??+

SLIDE 19

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

SLIDE 20

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

??-

SLIDE 21

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

SLIDE 22

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

??0

SLIDE 23

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

SLIDE 24

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

SLIDE 25

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

??-

SLIDE 26

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

SLIDE 27

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

??+

SLIDE 28

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

SLIDE 29

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1 4 7 10 2 5 8 11

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

??0

SLIDE 30

If the left side is heavier, record a +. If the right side is heavier, record a -. If the scale balances, record a 0.

SLIDE 31

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

SLIDE 32

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

2 3 4 11 12

SLIDE 33

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

2 3 4 11 12

SLIDE 34

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

2 3 4 11 5 6 7 12

SLIDE 35

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +0- +00 +0+ ++- ++0 12

2 3 4 11 5 6 7 12

+-- +-0 +-+

SLIDE 36

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +0- +00 +0+ ++- ++0 12

2 3 4 11 5 6 7 12

+-- +-0 +-+

SLIDE 37

Our Encoding Scheme

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

Lighter Heavier

SLIDE 38

Our Encoding Scheme

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

Lighter Heavier

SLIDE 39

Our Encoding Scheme

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+

Lighter Heavier

-0
12

SLIDE 40

Our Encoding Scheme

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+

Lighter Heavier

-0
12

SLIDE 41

Our Encoding Scheme

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+

Lighter Heavier

-0
12

SLIDE 42

Our Encoding Scheme

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
11

00- 0-+ 0-0 0--

++
+0
+-
-+

Lighter Heavier

-0
12
8
9
10
0+
00
0-

SLIDE 43

Our Encoding Scheme

1 2 3 4 5 6 7 11 00+ 0+- 0+0 0++ +-- +-0 +-+ ++- ++0 12

1
2
3
4
5
6
7
11

00- 0-+ 0-0 0--

++
+0
+-
-+

Lighter Heavier

-0
12

8 9 10 +0- +00 +0+

8
9
10
0+
00
0-

SLIDE 44

The Solution

5 6 7 11 8 9 10 12 2 3 4 11 5 6 7 12 1 4 7 8 2 5 10 11

1 2 3 4 5 6 7 11 00+ 0+- 0+0 0++ +-- +-0 +-+ ++-

-0
12
8
9
10
0+
00
0-

SLIDE 45

Generalizing the Result

Why twelve balls?
With three trits, 27 possible combinations. 13 are positive, nine

start with +.

Must discard one starting with + to ensure number of + and - in

each column is the same, leaving 12 positive numbers.

More generally:
With n trits, 3n possible combinations. (3n – 1) / 2 are positive.
3n-1 numbers start with +. To balance + and -, we need to drop
ne starting with +, leaving (3n – 3) / 2 positive numbers.
Can do 3, 12, 39, 120, 363, 1092, ...

SLIDE 46

1 2 3 0+ +- +0 4 ++

Example: Two Weighings

1

0-

2
+
4
3

Lighter Heavier

SLIDE 47

1 2 3 0+ +- +0 4 ++

Example: Two Weighings

1

0-

2
+
4
3

Lighter Heavier

SLIDE 48

1 2 3 0+ +- +0

Example: Two Weighings

1

0-

2
+
3

Lighter Heavier

SLIDE 49

1 2 3 0+ +- +0

Example: Two Weighings

1

0-

2
+
3

Lighter Heavier

SLIDE 50

1 2 3 0+ +- +0

Example: Two Weighings

1

0-

2
+
3

Lighter Heavier

SLIDE 51

1 2 3 0+ +- +0

Example: Two Weighings

1

0-

2
+
3

Lighter Heavier

SLIDE 52

Example: Three Weighings

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

Lighter Heavier +++ 13

13

SLIDE 53

Example: Three Weighings

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

Lighter Heavier +++ 13

13

SLIDE 54

Example: Three Weighings

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+
-0
12

Lighter Heavier

SLIDE 55

Example: Three Weighings

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++-

-0
12
1
2
3
4
5
6
7
8
9
10
11

00- 0-+ 0-0 0--

++
+0
+-
0+
00
0-
-+

Lighter Heavier ++0 12

SLIDE 56

Example: Three Weighings

1 2 3 4 5 6 7 8 9 10 11 00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0 12

1
2
3
4
5
6
7
11

00- 0-+ 0-0 0--

++
+0
+-
-+

Lighter Heavier

-0
12
8
9
10
0+
00
0-

SLIDE 57

Example: Three Weighings

1 2 3 4 5 6 7 11 00+ 0+- 0+0 0++ +-- +-0 +-+ ++- ++0 12

1
2
3
4
5
6
7
11

00- 0-+ 0-0 0--

++
+0
+-
-+

Lighter Heavier

-0
12

8 9 10 +0- +00 +0+

8
9
10
0+
00
0-

SLIDE 58

Example: Four Weighings

1 2 3 4 5 6 7 8 9 10 11 000+ 00+- 00+0 00++ 0+-- 0+-0 0+-+ 0+0- 0+00 0+0+ 0++- 0++0 12 0+++ 13 14 15 16 17 18 19 20 21 22 23 24 +--- +--0 +--+ +-0- +-00 +-0+ +-+- +-+0 +-++ +0-- +0-0 25 +0-+ 26 27 28 29 30 31 32 33 34 35 36 37 +000 +00+ +0+- +0+0 +0++ ++-- ++-0 ++-+ ++0- ++00 ++0+ +++- 38 +++0 39 +00- ++++ 40

SLIDE 59

Example: Four Weighings

1 2 3 4 5 6 7 8 9 10 11 000+ 00+- 00+0 00++ 0+-- 0+-0 0+-+ 0+0- 0+00 0+0+ 0++- 0++0 12 0+++ 13 14 15 16 17 18 19 20 21 22 23 24 +--- +--0 +--+ +-0- +-00 +-0+ +-+- +-+0 +-++ +0-- +0-0 25 +0-+ 26 27 28 29 30 31 32 33 34 35 36 37 +000 +00+ +0+- +0+0 +0++ ++-- ++-0 ++-+ ++0- ++00 ++0+ +++- 38 +++0 39 +00- ++++ 40

SLIDE 60

Example: Four Weighings

1 2 3 4 5 6 7 8 9 10 11 000+ 00+- 00+0 00++ 0+-- 0+-0 0+-+ 0+0- 0+00 0+0+ 0++- 0++0 12 0+++ 13 14 15 16 17 18 19 20 21 22 23 24 +--- +--0 +--+ +-0- +-00 +-0+ +-+- +-+0 +-++ +0-- +0-0 25 +0-+ 26 27 28 29 30 31 32 33 34 35 36 37 +000 +00+ +0+- +0+0 +0++ ++-- ++-0 ++-+ ++0- ++00 ++0+ +++- 38 +++0 39 +00-

SLIDE 61

Example: Four Weighings

1 2 3 4 5 6 7 8 9 10 11 000+ 00+- 00+0 00++ 0+-- 0+-0 0+-+ 0+0- 0+00 0+0+ 0++- 0++0 12 0+++ 13 14 15 16 17 18 19 20 21 22 23 24 +--- +--0 +--+ +-0- +-00 +-0+ +-+- +-+0 +-++ +0-- +0-0 25 +0-+ 26 27 28 29 30 31 32 33 34 35 36 37 +000 +00+ +0+- +0+0 +0++ ++-- ++-0 ++-+ ++0- ++00 ++0+ +++- 38 +++0 39 +00-

SLIDE 62

Example: Four Weighings

1 2 3 4 5 6 7 8 9 10 11 000+ 00+- 00+0 00++ 0+-- 0+-0 0+-+ 0+0- 0+00 0+0+ 0++- 0++0 12 0+++ 13 14 15 16 17 18 19 20 21 22

23
24

+--- +--0 +--+ +-0- +-00 +-0+ +-+- +-+0 +-++

0++
0+0
25
0+-
26
27
28
29
30
31

32 33 34

35
36
37
000
00-
0-+
0-0
0--

++-- ++-0 ++-+

-0+
-00
-0-

+++- 38

--0
39
00+

SLIDE 63

Example: Four Weighings

1 2 3 4 5 6 7 8 9 10 11 000+ 00+- 00+0 00++ 0+-- 0+-0 0+-+ 0+0- 0+00 0+0+ 0++- 0++0 12 0+++ 13 14 15 16 17 18 19 20 21 22

23
24

+--- +--0 +--+ +-0- +-00 +-0+ +-+- +-+0 +-++

0++
0+0
25
0+-
26
27
28
29
30
31

32 33 34

35
36
37
000
00-
0-+
0-0
0--

++-- ++-0 ++-+

-0+
-00
-0-

+++- 38

--0
39
00+

SLIDE 64

Some Insights

What number did we drop?
With 2 trits, dropped ++.
With 3 trits, dropped +++.
With 4 trits, dropped ++++.
Always drop ++…++
What numbers did we invert?
With two trits: +0
With three trits: ++0, +0-, +00, +0-
With four trits: +++0, ++0-, ++00, ++0+, +0--, +0-0, +0-+, +00-,

+000, +00+, +0+-, +0+0, +0++

Always invert numbers starting with ++…++0.
This always works!

SLIDE 65

How Many Extra +s Per Column?

Answer: The 3j column has (3j – 1) / 2 extra +'s.

0+ +- +0

SLIDE 66

How Many Extra +s Per Column?

Answer: The 3j column has (3j – 1) / 2 extra +'s.

00+ 0+- 0+0

SLIDE 67

How Many Extra +s Per Column?

Answer: The 3j column has (3j – 1) / 2 extra +'s.

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

SLIDE 68

How Many Extra +s Per Column?

Answer: The 3j column has (3j – 1) / 2 extra +'s.

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

SLIDE 69

How Many +'s Get Flipped?

Answer: The 3j column has (3j-1) / 2 flipped +'s.

SLIDE 70

How Many +'s Get Flipped?

Answer: The 3j column has (3j-1) / 2 flipped +'s.

0+ +- +0

SLIDE 71

How Many +'s Get Flipped?

Answer: The 3j column has (3j-1) / 2 flipped +'s.

0+ +- +0

SLIDE 72

How Many +'s Get Flipped?

Answer: The 3j column has (3j-1) / 2 flipped +'s.

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

SLIDE 73

How Many +'s Get Flipped?

Answer: The 3j column has (3j-1) / 2 flipped +'s.

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

SLIDE 74

How Many +'s Get Flipped?

Answer: The 3j column has (3j-1) / 2 flipped +'s.

00+ 0+- 0+0 0++ +-- +-0 +-+ +0- +00 +0+ ++- ++0

∑

i=0 j−1

3

i=3 j−1

2

SLIDE 75

Summary

A three-way scale lends itself naturally to a

balanced ternary encoding for each of the balls.

Given an encoding with the same number of +'s

and -'s in each column, we can use the scale to read off one trit of the answer at a time.

Flipping numbers starting with +0, ++0, etc.

guarantees an encoding with this property.

SLIDE 76

Generating Permutations

SLIDE 77

“You are given a sorted string S of unique

characters. Write a Java-style iterator that

traverses all the permutations of S in lexicographical order.”

SLIDE 78

Listing Lehmer Codes

1 2 3 4 5 6 7 8 9 10 11 2000 2010 2100 2110 2200 2210 3000 3010 3100 3110 3200 3210 abcd abdc acbd acdb adbc adcb bacd badc bcad bcda bdac bdac cabd cadb cbad cbda cdab cdba dabc dacb dbac dbca dcab dcba 12 13 14 15 16 17 18 19 20 21 22 23 0000 0010 0100 0110 0200 0210 1000 1010 1100 1110 1200 1210

SLIDE 125

Factoradic Numbers

Mixed-radix number system.
Nth digit in base n!.
Nth digit can be 0, 1, 2, …, n
Example: 3110!
3 x 3! + 1 x 2! + 1 x 1! + 0 x 0! = 21
Example: 1210!
1 x 3! + 2 x 2! + 1 x 1! + 0 x 0! = 11

SLIDE 126

Listing Lehmer Codes

1 2 3 4 5 6 7 8 9 10 11 2000 2010 2100 2110 2200 2210 3000 3010 3100 3110 3200 3210 abcd abdc acbd acdb adbc adcb bacd badc bcad bcda bdac bdac cabd cadb cbad cbda cdab cdba dabc dacb dbac dbca dcab dcba 12 13 14 15 16 17 18 19 20 21 22 23 0000 0010 0100 0110 0200 0210 1000 1010 1100 1110 1200 1210

SLIDE 127

Listing Lehmer Codes

1 2 3 4 5 6 7 8 9 10 11 2000 2010 2100 2110 2200 2210 3000 3010 3100 3110 3200 3210 abcd abdc acbd acdb adbc adcb bacd badc bcad bcda bdac bdac cabd cadb cbad cbda cdab cdba dabc dacb dbac dbca dcab dcba 12 13 14 15 16 17 18 19 20 21 22 23 0000 0010 0100 0110 0200 0210 1000 1010 1100 1110 1200 1210

SLIDE 128

Writing n in factoradic gives the nth Lehmer code.

SLIDE 129

Converting to Factoradic

Goal: convert k to factoradic.
Assume we know n, the number of elements to

permute.

To get the (n – 1)! place, divide k by (n – 1)!
Quotient is the (n – 1)! place.
Repeat for the remaining digits using the

remainder.

Identical to converting to any other base, just

using factorials instead of powers.

SLIDE 130

Example: Convert 13 to Factoradic

SLIDE 131

Example: Convert 13 to Factoradic

13 = 2 x 3! + 1

SLIDE 132

Example: Convert 13 to Factoradic

13 = 2 x 3! + 1 1 = 0 x 2! + 1

SLIDE 133

Example: Convert 13 to Factoradic

13 = 2 x 3! + 1 1 = 0 x 2! + 1 1 = 1 x 1! + 0

SLIDE 134

Example: Convert 13 to Factoradic

13 = 2 x 3! + 1 1 = 0 x 2! + 1 1 = 1 x 1! + 0 0 = 0 x 0! + 0

SLIDE 135

Example: Convert 13 to Factoradic

13 = 2 x 3! + 1 1 = 0 x 2! + 1 1 = 1 x 1! + 0 0 = 0 x 0! + 0

SLIDE 136

Example: Convert 13 to Factoradic

13 = 2 x 3! + 1 1 = 0 x 2! + 1 1 = 1 x 1! + 0 0 = 0 x 0! + 0 Answer: 2010!

SLIDE 137

Generating Permutations

public static String kthPermutation(String chars, int k) { String result = ""; for (int n = chars.length() - 1; n >= 0; --n) { int quotient = k / factorial(n); int remainder = k % factorial(n); result += chars.charAt(quotient); chars = chars.substring(0, quotient) + chars.substring(quotient + 1); k = remainder; } return result; }

SLIDE 138

Analysis of Our Algorithm

To get the kth permutation of n elements:
Converting k to factoradic takes O(n)
Building up the permutations takes O(n2 )

– Elements stored in a list; O(n) to remove each.

Can reduce to O(n lg n) using order statistic tree.
Easy to build an iterator from this.

Incrementing Factoradic Numbers

4 1

SLIDE 160

Incrementing Factoradic Numbers

Find the digit to increment.
Scan backwards from the end to find the first

number not at its maximum.

Increment that digit.
Set the digits after that to zero.

Incrementing Permutations

4 1

SLIDE 192

4 A C D B

Incrementing Permutations

4 1

SLIDE 193

Incrementing Permutations

Find the digit to be incremented.
Scan backwards from the end of the sequence to find

the longest increasing sequence.

Increment that digit.
Find the smallest element bigger than the element right

before that sequence.

Swap those two elements.
Set the digits after that to zero.
Reverse the increasing sequence
Runtime: O(n) per permutation.

SLIDE 194

Summary

Lehmer codes describe a permutation as a

series of elements to choose in order.

The factoradic number system maps directly
nto Lehmer codes, and thus onto

permutations.

By simulating what would happen if we wrote
ut the Lehmer code, we can derive a fast

algorithm for generating ordered permutations.

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Concluding Thoughts

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Number systems are useful in number recovery by discovering one component of the number at a time.

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Number systems are useful in enumeration by revealing structure hidden in the indices.

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Fun With Number Systems

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